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        <title>MFEM - Finite Element Discretization Library</title>

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    <style type="text/css">.MathJax_Hover_Frame {border-radius: .25em; -webkit-border-radius: .25em; -moz-border-radius: .25em; -khtml-border-radius: .25em; box-shadow: 0px 0px 15px #83A; -webkit-box-shadow: 0px 0px 15px #83A; -moz-box-shadow: 0px 0px 15px #83A; -khtml-box-shadow: 0px 0px 15px #83A; border: 1px solid #A6D ! important; display: inline-block; position: absolute}
.MathJax_Menu_Button .MathJax_Hover_Arrow {position: absolute; cursor: pointer; display: inline-block; border: 2px solid #AAA; border-radius: 4px; -webkit-border-radius: 4px; -moz-border-radius: 4px; -khtml-border-radius: 4px; font-family: 'Courier New',Courier; font-size: 9px; color: #F0F0F0}
.MathJax_Menu_Button .MathJax_Hover_Arrow span {display: block; background-color: #AAA; border: 1px solid; border-radius: 3px; line-height: 0; padding: 4px}
.MathJax_Hover_Arrow:hover {color: white!important; border: 2px solid #CCC!important}
.MathJax_Hover_Arrow:hover span {background-color: #CCC!important}
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</style></head>

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        <li class="main active"><a href="#tensor-based-operator-assembly-and-evaluation">Tensor-Based Operator Assembly and Evaluation</a></li>
        
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<h1 id="tensor-based-operator-assembly-and-evaluation">Tensor-Based Operator Assembly and Evaluation</h1>
<h2 id="overview">Overview</h2>
<p>The high performance (HPC) versions of the example codes in the <code>miniapps/performance</code> directory
use a set of templated classes to efficiently implement the inner-most portion
(<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msup&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.842ex" height="2.378ex" viewBox="0 -908.4 2945.7 1023.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="1074" y="513"></use><use xlink:href="#MJMATHI-44" x="1357" y="0"></use><use xlink:href="#MJMATHI-42" x="2186" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>B</mi><mi>T</mi></msup><mi>D</mi><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-1">B^T D B</script>) of the fundamental finite element (FE) operator decomposition:</p>
<p><img alt="" src="../../doc/web/FEDecomposition.png"></p>
<p>We take advantage of the tensor-product structure of both the finite element
basis and the quadrature rule to efficiently apply the action of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.115ex" viewBox="0 -795.1 759.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-2">B</script> without
necessarily computing its entries. This is generally know as <em><a href="http://www.sciencedirect.com/science/article/pii/0021999180900054">sum
factorization</a></em>.
In the case where we pre-compute and store the <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-3">D</script> matrix, we call the algorithm
<em>partial assembly</em>.</p>
<h2 id="template-implementation">Template implementation</h2>
<p>Below is a short description of the header files that are part of the initial
reference implementation of the tensor-based finite element assembly and
evaluation algorithms.</p>
<h3 id="configtconfighpp">config/tconfig.hpp</h3>
<p>Configuration macros including the specification of the template blocking
   size (currently used in the matrix-matrix multiply).</p>
<h3 id="generaltassignhpp">general/tassign.hpp</h3>
<p>Helper templated function to allow single implementation of multiple
   assignment operations on tensor entries.</p>
<h3 id="linalgtlayouthpp">linalg/tlayout.hpp</h3>
<p>Classes describing fixed size tensor layouts. Implemented are standard
   strided layouts for 1D/2D/3D/4D tensors. Layouts support reshape and
   sub-tensor operations independent of the actual data storage.</p>
<p>This header also contains another set of "vector layout" classes for
   converting scalar data indices into multi-component (vector) data indices.
   They are used to describe the layout of vector <code>GridFunctions</code> on global
   degrees of freedom (similar to the <code>Ordering</code> class). In the FE operator
   decomposition, these classes are used by the templated <code>*_FiniteElementSpace</code>
   classes (see <code>fem/tfespace.hpp</code> below) to implement the actions of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.827ex" height="2.115ex" viewBox="0 -795.1 786.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-47" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script type="math/tex" id="MathJax-Element-4">G</script> and
   <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.216ex" height="2.378ex" viewBox="0 -908.4 1384.7 1023.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-47" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="1112" y="513"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>G</mi><mi>T</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-5">G^T</script> in the case of vector (multi-component) input and/or output fields.</p>
<h3 id="linalgtmatrixhpp">linalg/tmatrix.hpp</h3>
<p>Small matrix operations, defined by specializations: determinant, adjugate,
   etc. Matrix-matrix multiply, <code>C = A.B</code>, simple and blocked version.</p>
<h3 id="linalgttensorhpp">linalg/ttensor.hpp</h3>
<p>Fixed-size container classes <code>TVector</code>, <code>TMatrix</code>, <code>TTensor3</code>, <code>TTensor4</code> for
   1D/2D/3D/4D tensors stored in column-major layout.
   Element-wise tensor operations: <code>A {=,+=,*=} scalar</code>; <code>A {=,+=,*=} B</code>.  The
   latter allows different input and output layouts. With suitable layouts this
   can be used to permute (transpose) tensors, extract sub-tensors, etc.
   The tensor contraction/product operations:</p>
<ul>
<li><code>Mult_1_2</code>: <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-6-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/munder&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="20.84ex" height="2.904ex" viewBox="0 -851.8 8972.8 1250.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-43" x="0" y="0"></use><g transform="translate(715,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1036" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="1315" y="0"></use></g><use xlink:href="#MJMAIN-3D" x="2391" y="0"></use><g transform="translate(3448,0)"><use xlink:href="#MJSZ1-2211" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="1494" y="-405"></use></g><g transform="translate(5103,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="469" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="748" y="0"></use></g></g><g transform="translate(6774,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1093" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="1372" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>=</mo><munder><mo>∑</mo><mi>s</mi></munder><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-6">C_{i,j,k} = \sum_s A_{s,j} B_{i,s,k}</script></li>
<li><code>Mult_2_1</code>: <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/munder&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="20.84ex" height="2.904ex" viewBox="0 -851.8 8972.8 1250.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-43" x="0" y="0"></use><g transform="translate(715,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1036" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="1315" y="0"></use></g><use xlink:href="#MJMAIN-3D" x="2391" y="0"></use><g transform="translate(3448,0)"><use xlink:href="#MJSZ1-2211" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="1494" y="-405"></use></g><g transform="translate(5103,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="624" y="0"></use></g></g><g transform="translate(6727,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="469" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="748" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1160" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="1439" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>=</mo><munder><mo>∑</mo><mi>s</mi></munder><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-7">C_{i,j,k} = \sum_s A_{i,s} B_{s,j,k}</script></li>
<li><code>TensorAssemble</code>: <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/munder&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="26.643ex" height="2.904ex" viewBox="0 -851.8 11471.3 1250.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-43" x="0" y="0"></use><g transform="translate(715,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1145" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="1424" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1836" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6C" x="2115" y="0"></use></g><use xlink:href="#MJMAIN-3D" x="2799" y="0"></use><g transform="translate(3856,0)"><use xlink:href="#MJSZ1-2211" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="1494" y="-405"></use></g><g transform="translate(5511,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="469" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="748" y="0"></use></g></g><g transform="translate(7135,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="469" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="748" y="0"></use></g></g><g transform="translate(8806,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="521" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="800" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1269" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6C" x="1547" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>=</mo><munder><mo>∑</mo><mi>s</mi></munder><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>i</mi></mrow></msub><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>l</mi></mrow></msub><mtext>&nbsp;</mtext><mtext>&nbsp;</mtext></math></span></span><script type="math/tex" id="MathJax-Element-8">C_{i,k,j,l} = \sum_s A_{s,i} A_{s,j} B_{k,s,l}~~</script> and
     <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;msub&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/munder&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="26.824ex" height="2.904ex" viewBox="0 -851.8 11549.3 1250.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(500,0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use><g transform="translate(828,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1145" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="1424" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1836" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6C" x="2115" y="0"></use></g></g><use xlink:href="#MJMAIN-3D" x="3412" y="0"></use><g transform="translate(4469,0)"><use xlink:href="#MJSZ1-2211" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="1494" y="-405"></use></g><g transform="translate(6124,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="624" y="0"></use></g></g><g transform="translate(7748,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="469" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="748" y="0"></use></g></g><g transform="translate(9428,0)"><use xlink:href="#MJMATHI-43" x="0" y="0"></use><g transform="translate(715,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="521" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-73" x="800" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1269" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6C" x="1547" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>&nbsp;</mtext><mtext>&nbsp;</mtext><msub><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>=</mo><munder><mo>∑</mo><mi>s</mi></munder><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mo>,</mo><mi>j</mi></mrow></msub><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>l</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-9">~~D_{i,k,j,l} = \sum_s A_{i,s} B_{s,j} C_{k,s,l}</script></li>
<li><code>TensorProduct</code>: <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.569ex" height="2.772ex" viewBox="0 -795.1 7564.4 1193.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-43" x="0" y="0"></use><g transform="translate(715,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1036" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="1315" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1836" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6C" x="2115" y="0"></use></g><use xlink:href="#MJMAIN-3D" x="2799" y="0"></use><g transform="translate(3856,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><g transform="translate(750,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="624" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="1036" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6B" x="1315" y="0"></use></g></g><g transform="translate(6005,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="412" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6C" x="691" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>=</mo><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-10">C_{i,j,k,l} = A_{i,j,k} B_{j,l}</script></li>
</ul>
<h3 id="meshtmeshhpp">mesh/tmesh.hpp</h3>
<p>The Mesh object templated by the finite element space and layout of the
   nodes. Provides <code>MatchesGeometry()</code> and <code>MatchesNodes()</code> functions to verify
   if compiled and runtime mesh agree.</p>
<h3 id="femtintruleshpp">fem/tintrules.hpp</h3>
<p>Quadrature rules templated by geometry (triangles, quads, etc.) and
   integration order.</p>
<h3 id="femtfehpp">fem/tfe.hpp</h3>
<p>H1 and L2 finite elements templated by geometry and polynomial order.</p>
<h3 id="femtfespacehpp">fem/tfespace.hpp</h3>
<p>Template <code>*_FiniteElementSpace</code> classes providing the mappings between global
   and local (element) degrees of freedom for H1 continuous and L2 discontinuous
   spaces. In the FE operator decomposition, these classes provide the element
   local action of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.827ex" height="2.115ex" viewBox="0 -795.1 786.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-47" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script type="math/tex" id="MathJax-Element-11">G</script> (<code>Extract</code> methods) and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.216ex" height="2.378ex" viewBox="0 -908.4 1384.7 1023.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-47" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="1112" y="513"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>G</mi><mi>T</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-12">G^T</script> (<code>Assemble</code> methods).</p>
<h3 id="femtcoefficienthpp">fem/tcoefficient.hpp</h3>
<p>Templated versions of classes derived from the abstract class <code>Coefficient</code>.
   It encapsulates physical quantities like material properties, sources,
   boundary/initial conditions, etc. Its main functionality is to evaluate the
   coefficient at all quadrature points in an element, which is then used in the
   evaluation of the <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-13">D</script> matrix.</p>
<h3 id="femteltranshpp">fem/teltrans.hpp</h3>
<p>Element transformation class, templated on a mesh type and an integration
   rule. It is constructed from a mesh (e.g. class <code>TMesh</code>) and shape evaluator
   (e.g. class <code>ShapeEvaluator</code>) objects. Allows computation of physical
   coordinates and Jacobian matrices corresponding to the reference integration
   points. The desired result (a combination of coordinates and/or Jacobian
   matrices at quadrature points, element attribute and/or element index) is
   specified through the template sub-class <code>Result</code> and stored in an object of
   the same type. The idea of this approach is to eliminate unnecessary
   evaluations if they are not needed. The need is determined based on what the
   particular "users" need. The "users" are the templated <code>Coefficient</code> and
   <code>Kernel</code> (see <code>fem/tbilininteg.hpp</code> below) classes which specify what they
   need through static constant boolean variables, e.g. <code>uses_coordinates</code>,
   <code>uses_Jacobians</code>, etc.</p>
<h3 id="femtevaluatorhpp">fem/tevaluator.hpp</h3>
<p>Classes for evaluating FE basis, <code>ShapeEvaluator</code>, and finite element
   functions, <code>FieldEvaluator</code>, and their derivatives at quadrature points,
   templated by a finite element class and an integration rule class. These
   correspond to the <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.115ex" viewBox="0 -795.1 759.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-14">B</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.591ex" height="2.115ex" viewBox="0 -795.1 1546 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMATHI-47" x="759" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>G</mi></math></span></span><script type="math/tex" id="MathJax-Element-15">BG</script> matrices above. Quads and hexes use the
   tensor-product structure for fast evaluation.</p>
<h3 id="femtbilininteghpp">fem/tbilininteg.hpp</h3>
<p><code>Kernel</code> classes (e.g. mass, diffusion) that represent the matrix <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-16-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-16">D</script> from
   the above FE operator decomposition. These classes also specify the <em>type</em> of
   the local operator that needs to be applied before and after the <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-17-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-17">D</script> matrix -
   these are the <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.55ex" height="2.378ex" viewBox="0 -795.1 1528.4 1023.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.531ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6E" x="345" y="0"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-18">B_{in}</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msubsup&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.328ex" height="3.036ex" viewBox="0 -908.4 1863.2 1307" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="1074" y="488"></use><g transform="translate(759,-279)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6F" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-75" x="485" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="1058" y="0"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>o</mi><mi>u</mi><mi>t</mi></mrow><mi>T</mi></msubsup></math></span></span><script type="math/tex" id="MathJax-Element-19">B^T_{out}</script> matrices, respectively. The product
   <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msubsup&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msubsup&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.802ex" height="3.036ex" viewBox="0 -908.4 4220.2 1307" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.926ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="1074" y="488"></use><g transform="translate(759,-279)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6F" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-75" x="485" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="1058" y="0"></use></g><use xlink:href="#MJMATHI-44" x="1863" y="0"></use><g transform="translate(2691,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><g transform="translate(759,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6E" x="345" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>o</mi><mi>u</mi><mi>t</mi></mrow><mi>T</mi></msubsup><mi>D</mi><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-20">B^T_{out} D B_{in}</script> is the local element matrix, which is the result when
   using the <code>BilinearFormIntegrator</code> classes. This specifications of the types
   are given by static constant boolean variables, e.g. <code>in_values</code> and
   <code>out_values</code>. The <code>Kernel</code> classes provide the following methods:</p>
<ul>
<li><code>Action</code>: evaluate the action of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-21">D</script> <em>without</em> explicitly storing the
     partially assembled data; this is needed for matrix-free action.</li>
<li><code>Assemble</code>: evaluate the partially assembled data, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-22-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-22">D</script>, which is
     kernel-specific: e.g., for mass, the data is one scalar per quadrature
     point; for diffusion, the data is one <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-23-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.272ex" height="2.115ex" viewBox="0 -795.1 2269.9 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use><use xlink:href="#MJMAIN-D7" x="745" y="0"></use><use xlink:href="#MJMATHI-64" x="1746" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>×</mo><mi>d</mi></math></span></span><script type="math/tex" id="MathJax-Element-23">d\times d</script> matrix (in <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-24-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.216ex" height="2.115ex" viewBox="0 -795.1 523.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script type="math/tex" id="MathJax-Element-24">d</script>-dimensions)
     per quadrature point.</li>
<li><code>MultAssembled</code>: perform the action of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-25-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.115ex" viewBox="0 -795.1 828.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-25">D</script> using the pre-computed partially
     assembled data.</li>
</ul>
<h3 id="femtbilinearformhpp">fem/tbilinearform.hpp</h3>
<p>Bilinear form operator, templated on the mesh, finite element space,
   integration rule and bilinear form integrator. Corresponds to the <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-26-Frame" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.743ex" height="2.115ex" viewBox="0 -795.1 750.5 910.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.268ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-26">A</script> matrix
   above. Provides various assembly and evaluation schemes:</p>
<ul>
<li><code>MultUnassembled</code>: matrix-free action using the mesh nodes and the input
     vector.</li>
<li><code>Assemble</code>, <code>MultAssembled</code>: partial assembly and operator action using the
     partially assembled data at quadrature points and the input vector.</li>
<li><code>AssembleMatrix(DenseTensor &amp;)</code>: assemble the local element matrices and
     store them as <code>DenseTensor</code>.</li>
<li><code>AssembleMatrix(SparseMatrix &amp;)</code>: assemble the operator in a global (CSR)
     <code>SparseMatrix</code>.</li>
<li><code>AssembleBilinearForm(BilinearForm &amp;)</code>: assemble element matrices and add
     them to the bilinear form.</li>
</ul>
<h3 id="miniappsperformancemakefile">miniapps/performance/makefile</h3>
<p>By default <code>make</code> builds the example drivers with the compiler used to
   compile MFEM. If <code>g++</code> was used, a pseudo-code dump file with the optimized
   code will be generated (option <code>-fdump-tree-optimized-blocks</code>). The <code>g++</code>
   option <code>--param max-completely-peel-times=3</code> prevents the compiler from
   unrolling innermost loops (of size greater than 3), allowing the compiler to
   vectorize them. Some options for optimization/vectorization with the clang
   compiler are also included.</p>
<h3 id="miniappsperformanceexcpp">miniapps/performance/ex*.cpp</h3>
<p>High-performance templated versions of the corresponding <code>examples/ex*.cpp</code>
   example codes.</p>
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